Our approach applies to any multitype branching process model under the standard rate-linearity assumption, enabling us to calculate a wide variety of pertinent transition probabilities, expectations and higher moments toward exploring the transient behavior of the particles. We see that the framework can be used to answer questions about the population of any given type \(X_i(t)\)  in a general multitype process. Similarly, one can pose related questions related to sums or differences of populations, such as the probability that one population becomes dominant or outnumbers another [1]}. In some cases, we can establish connections to related models and results in the literature. For instance, an anonymous reviewer remarks that we may recognize the equation corresponding to the final type as the birth-death equations, for which additional tools are available. Indeed, if one investigates the probability of a double-mutant in the three stage model depicted in Figure 2, the problem can be equivalently case in terms of survival probabilities of birth-death processes can be used [2]}, [3]}.  Exploring further connections and possible extensions of these techniques to systems with non-linear interaction dynamics will be a fruitful avenue for future work.
